For real values of parameter and positive argument , the values of the exponential integral are real (or infinity). For real values of argument , the values of the exponential integral , the sine integral , and the hyperbolic sine integral are real. For real positive values of argument , the values of the logarithmic integral , the cosine integral , and the hyperbolic cosine integral are real.

The exponential integrals have rather simple values for argument :

If the parameter equals , the exponential integral can be expressed through an exponential function multiplied by a simple rational function. If the parameter equals , the exponential integral can be expressed through the exponential integral , and the exponential and logarithmic functions:

The previous formulas are the particular cases of the following general formula:

If the parameter equals , the exponential integral can be expressed through the probability integral , and the exponential and power functions, for example:

The previous formulas can be generalized by the following general representation of this class of particular cases:

The exponential integrals , , , , , , and are defined for all complex values of the parameter and the variable . The function is an analytical functions of and over the whole complex ‐ and ‐planes excluding the branch cut on the ‐plane. For fixed , the exponential integral is an entire function of . The sine integral and the hyperbolic sine integral are entire functions of .

For fixed , the function has an essential singularity at . At the same time, the point is a branch point for generic . For fixed , the function has only one singular point at . It is an essential singular point.

The exponential integral , the cosine integral , and the hyperbolic cosine integral have an essential singularity at .

The function does not have poles and essential singularities.

The sine integral and the hyperbolic sine integral have an essential singularity at .

For fixed , the function does not have branch points and branch cuts.

For fixed , not being a nonpositive integer, the function has two branch points and , and branch cuts along the interval . At the same time, the point is an essential singularity for this function.

The exponential integral , the cosine integral, and the hyperbolic cosine integral have two branch points and .

The function has three branch points , , and .

The sine integral and hyperbolic sine integral do not have branch points or branch cuts.

For fixed , not being a nonpositive integer, the function is a single‐valued function on the ‐plane cut along the interval , where it is continuous from above:

The function is a single‐valued function on the ‐plane cut along the interval , where it has discontinuities from both sides:

The function is a single‐valued function on the ‐plane cut along the interval . It is continuous from above along the interval and it has discontinuities from both sides along the interval :

The cosine integral and hyperbolic cosine integral are single‐valued functions on the ‐plane cut along the interval where they are continuous from above:

From below, these functions have discontinuity that are described by the formulas:

The exponential integrals , , , , , , and do not have periodicity.

The exponential integral has mirror symmetry:

The sine integral and the hyperbolic sine integral are odd functions and have mirror symmetry: